A right triangle has leg lengths of
Guidance
In addition to using crossmultiplication to solve a rational equation, we can also use the LCD of all the rational expressions within the equation and eliminate the fraction. To demonstrate, we will walk through a few examples.
Example A
Solve
Solution:
The LCD for 2 and
Checking the answer, we have
Example B
Solve
Solution:
Because the denominators are the same, we need to multiply all three terms by
Checking our answer, we have:
Example C
Solve
Solution:
Determine the LCD for 5,
Multiplying each term by the entire LCD cancels out each denominator, so that we have an equation that we have learned how to solve in previous concepts. Distribute the 15 and
This polynomial is not factorable. Let’s use the Quadratic Formula to find the solutions.
Approximately, the solutions are
The
Intro Problem Revisit We need to use the Pythagorean Theorem to solve for x .
Guided Practice
Solve the following equations.
1.
2.
3.
Answers
1. The LCD is
Checking our answer, we have:
2. The LCD is
This polynomial factors to be
3. The LCD is
This quadratic is not factorable, so we need to use the Quadratic Formula to solve for
Using your graphing calculator, you can check the answer. The
Explore More
Determine if the following values for x are solutions for the given equations.

4x−3+2=3x+4, x=−1 
2x−1x−5−3=x+62x, x=6
What is the LCD for each set of numbers?

4−x, x2−16 
2x, 6x−12, x2−9 
x−3, x2−x−6, x2−4
Solve the following equations.

6x+2+1=5x 
53x−2x+1=4x 
12x2−9=8xx−3−2x+3 
6xx2−1+2x+1=3xx−1 
5x−34x−x+1x+2=1x2+2x 
4xx2+6x+9−2x+3=3x2−9 
x2x2−8x+16=xx−4+3xx2−16 
5x2x−3+x+1x=6x2+x+122x2−3x 
3xx2+2x−8=x+1x2+4x+2x+1x2−2x 
x+1x2+7x+x+2x2−3x=xx2+4x−21